ES2339295T3 - DECODIFICATION PROCESS IN MESSAGE PASS WITH A CLASSIFICATION OF THE NODES ACCORDING TO A RELIABILITY RELIABILITY. - Google Patents

Abstract

The invention relates to an iterative method by message passing for decoding of an error correction code that can be displayed in a bipartite graph comprising a plurality of variable nodes and a plurality of check nodes. For each iteration in a plurality of decoding iterations of said method: variable nodes or check nodes are classified (720) as a function of the corresponding degrees of reliability of decoding information available in the neighborhoods (Vn(d),Vm(d)) of these nodes, a node with a high degree of reliability being classified before a node with a low degree of reliability; each node thus classified (725) passes at least one message (αmn,βmn) to an adjacent node, in the order defined by said classification. The invention also relates to a computer program designed to implement said decoding method.

Description

Decoding procedure in step of messages with a classification of the nodes according to a reliability of neighborhood.

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Technical field

The present invention relates to the decoding of error correction codes in the field of telecommunications and data recording. More precisely, the invention is related to an iterative decoding process in the passage of error correction code messages susceptible to a bipartite graphic representation, such as LDPC ( Low Density Parity Check ) codes or turbocodes.

Prior art

Corrective error codes of representation by means of a bipartite graph cover a large code diversity, particularly LDPC codes, initially described by R. Gallager in his article entitled "Low density parity check codes ", published in IEEE Trans. Inform. Theory, vol. IT-8, pages 21-28, 1962, whose interesting properties have been rediscovered recently, and the turbocodes introduced by C. Berrou and cols., in its founding article "Near optimum error correcting coding and decoding ": turbocodes, appeared in IEEE Trans. Inform. Theory, vol. 44, no. 10, pages 1261-1271, nineteen ninety six.

It is called a two-part graph a graph not oriented, whose set of nodes consists of two disjoint subsets such as any two nodes of a same subset that are not linked together by an edge of the graphic.

Some error correction codes are susceptible of a representation by means of a bipartite graph. He chart is divided into a first subset of nodes associated with the symbols that constitute a code word, and a second subset of nodes associated with code restrictions, typically at parity controls. A bipartite chart associated with a group of restrictions, it is also called a graph of Tanner

The symbols of the code word are, in general, elements of the body of Galois F_ {2} = {0,1}, in other words of the bits, but they can be, more generally, elements of a body of feature 2 any F 2 ' , and therefore of a 2 p -ary alphabet. In what follows, we will limit ourselves without loss of generality to the case in which
p = 1, that is, to binary codes.

The codes that can be represented by a two-part graphic can be decoded by means of an iterative decoding in message passing, also known as MP ( Message Passing ) or BP ( Belief Propagation ). A generic description of this method of decoding can be found in the thesis of N. Wiberg, entitled "Codes and decoding on general graphs", 1996. The iterative decoding of type MP is, in fact, a generalization of well-known algorithms in the field of decoding, namely the "forward-backward" algorithm used for turbo codes, and the Gallager algorithm for LDPC codes.

In an attempt to simplify, we will resort to what follows the principle of iterative decoding by passing messages within the framework of an LDPC code. We will consider a linear code { K, N } where K is the dimension of the code that represents the number of bits of information, and N is the length of the code, which represents the number of coded bits. M = N - K corresponds to the number of parity bits or, equivalently, to the number of parity restrictions.

The Tanner graph of a linear code { K, N } is illustrated in Figure 1. The nodes corresponding to the bits of the code, also called "variable" or even more simply "variable" types, and to the right, the nodes corresponding to the parity controls have been included on the left of the graph. , also called nodes of type "control", or even more simply "controls". The incidence matrix of the graph corresponds to the parity matrix of the code that is of dimension M x N. Thus, the bipartite graph comprises N nodes of type "variable" and M nodes of type "control", being a node n of variable linked to a control node m if, and only if,
h_ {nm} = 1. For example, the graph in Figure 1 corresponds to a code (10.5) that has the following parity matrix:

one

It is generally recalled that a linear code is defined by a generating matrix G whose elements are binary linear values, and that a code word x = (x_ {1}, x_ {2}, ..., x_ { N}) is obtained from a word of information bits a = (\ alpha_ {1}, \ alpha_ {2}, ..., \ alpha_ {k}) by means of:

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100

Since all code words Check the parity controls, you have the relationship:

101

where indicated by G ^ {T} transform matrix G.

The code word x is transmitted by a communication channel, or registered in a data carrier. With the reception or reading of the medium, a noisy version of x is recovered, namely y = ( y_ {1}, y_ {2}, ..., y_ {N} ). The decoding operation consists in finding xy with it a , from the observation y .

We will agree the following notations before describe the principle of iterative decoding in step of posts:

H (n) designates the set of controls linked to variable n in the bipartite graph, or in other words, the set of nodes adjacent to node n ;

H (m) is the set of variables associated with the control m in the bipartite graph, or in other words, the set of nodes adjacent to the node m ;

α_ {n} represents the a priori information concerning the variable n of the bipartite graph, or in other words, the a priori information concerning the nth bit of the code word. This information takes into account the received signal and the characteristics of the transmission channel. It constitutes the decoder input, and is generally supplied by the demodulator in the form of flexible values, that is to say in terms of probabilities:

102

where p_ {n} ^ {\ alpha} = Pr (x_ {n} = \ alpha | y_ {n} , \ alpha \ in {0,1},

that is, more comfortably in the form of a logarithmic ratio of probabilities or LLR:

2

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That way, using a BBCG noise (noise Gaussian centered white) and a BPSK modulation, the demodulator simply calculate:

3

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 where \ sigma2 is the variance of noise.

α_ {mn} represents the message transmitted by the variable n to the control m \ in H ( n ). By reference to the turbo codes, α_ {mn} is even known as extrinsic information;

βnm represents, symmetrically, the message transmitted by the control m to the variable n \ in H ( m ). It is also classified as extrinsic information;

\ hat {α} n represents the a posteriori information relative to the variable n ; it takes into account both the a priori \ alpha_ {n} information and the \ beta_ {nm} messages received by the variable n from its adjacent controls during decoding;

\ overline {\ alpha} _ {n} is the rigid value corresponding to the flexible value \ hat {\ alpha} _ {n}, in other words the sampling for bit x_ {n} .

The principle of iterative decoding by Message passing has been illustrated in Figure 2.

In step 210, the initialization of the messages \ alpha_ {mn} is carried out , for each pair of variable n and control m \ in H ( n ). The messages \ alpha_ {mn} are generally initialized by the a priori information, that is: \ alpha_ {mn} = \ alpha_ {n} , \ forall m \ in H ( n ). The Iter iteration counter is also initialized by setting to 0.

The initialization stage is followed by a iteration loop comprising the following stages:

In 220, the controls are processed. More precisely, for each control m , the messages β of the control m are calculated with destination to the respective variables n \ in H ( m ), that is:

4

in which the control treatment function has been indicated with F_ {c} . For a pair of nodes m, n \ in H (m) any given message \ beta_ {nm} is calculated based on the messages itself received the m control from the variables n '\ in H (m) - { n }. It is found, therefore, that there is no forwarding of extrinsic information from a variable node on itself. The control treatment stage is also called the horizontal stage.

In 230, we proceed symmetrically to the treatment of the variables. More precisely, for each variable n , the messages α_ {mn} are calculated for the respective controls m \ in H ( n ), that is:

5

in which the treatment function of the variables has been indicated with F_ {v} . For a pair of nodes n, m \ in H ( n ) given, the message \ alpha_ {nm} is calculated based on the messages that have received the variable n of the controls m ' \ in H ( n ) - { m } So, as in the foregoing, there is no forwarding of extrinsic information from a node itself. The stage of treatment of the variables is also called the vertical stage.

At 240, the a posteriori information \ hat {\ alpha} _ {n} is estimated from the a priori information \ alpha_ {n} and the messages \ beta_ {mn} received by the variable n from their nodes adjacent control m \ in H ( n ), which is expressed symbolically by:

6

in which the a posteriori estimation function has been indicated with F_ {AP} .

In 250, a decision is made about the values rigid \ overline {\ alpha} _ {n} from flexible values \ hat {\ alpha} n, that is:

7

in which the decision function has been indicated with F_ {0} . Typically, for a BPSK modulation, the decision is made on the sign of the flexible value, that is, \ overline {\ alpha} _ {n} = sgn (\ hat {\ alpha} n). The value of a bit with its modulated value will be identified in the following, for convenience. The "0" bit will be conventionally represented by the "+1" value and the "1" bit by the "-1" value.

In 260, it is checked whether the vector \ overline {\ alpha} = ( \ alpha_ {1}, \ alpha_ {3}, ..., \ alpha_ {N} ) is a code word, that is, if it satisfies the parity controls If so, the loop is exited at 265, with \ overline {\ alpha} being the decoded word. If not, the number of iterations is increased by 267, and it is compared, at 270, if the number of iterations carried out Iter has reached a threshold value Iter _max. If that is not the case, iterative decoding is continued by returning to the loop of step 220. Failing that, the failure of the decoding operation is concluded, and exit 275.

The order of the stages in the iteration loop may differ from that shown in Figure 2. In particular, the order of processing of the variables and controls can be reversed, and then begin with an initialization of the messages \ beta_ {mn} : \ beta_ {mn} = 0, \ forall n \ in {1, ..., N } and \ forall m \ in H ( n ).

According to the principle of iterative decoding presented in Figure 2, we proceed to the treatment of all controls and after all the variables, or, according to mentioned above, to the treatment of all variables and after all controls. There is talk about "parallel planning", or even "planning by flooding (scheduling). Other types of planning they have been proposed in the bibliography; these might be classified into two categories:

- serial type planning, category in that the so-called "serial" schedules can be sorted scheduling "," shuffled-BP "," horizontal shuffled "or" vertical shuffled. "Type planning series is also applicable to both controls and variables For an application to controls, decoding use the following strategy:

?

first, we proceed to the treatment of a single threshold m by calculating the messages \ beta_ {mn} destined to the variables n \ in H ( m );

?

the messages \ alpha_ {mn} of each variable n \ in H ( m ) are updated and transmitted to the controls m ' \ in H ( n ) - { m }. The information a posteriori \ hat {\ n}, relative to the same variables, is also updated;

?

the treatment of next check and the previous two steps are repeated until the depletion of controls.

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In a dual way, you can make a treatment, variable by variable, instead of making a control by control treatment. According to the case provided, we will talk "horizontal shuffled" planning or planning "vertical shuffled".

You can also hybridize both schedules mentioned above in the form of a schedule called "mixed" or "group-shuffled". Be you will find a description of the decoding strategy corresponding to a mixed planning in the J- article Zhang et al., Entitled "Shuffled iterative decoding" appeared  in IEEE Trans. On Comm., Vol. 53, no. 2, February 2005, pages 209-213. The strategy is based on a partition of the nodes by groups, being the parallel treatment in the within a group, and in series from one group to the next. By way of more precise, for a partition in groups of controls:

?

the first group G is formed by the controls { m_ {1}, m_ {2}, ..., m_ {g} } calculating the messages \ beta_ {mn} destined to the variables n \ in H ( m_ {i} ) for i = 1, ..., g ;

?

the messages \ alpha_ {mn} of each variable n \ in H ( m_ {i} ), i = 1, ..., g are updated and transmitted to the respective controls m ' \ in H ( n ) - { m }. The a posteriori \ α {n} information related to these same variables is also updated;

?

the treatment of next group of controls, and the previous two stages are repeated until the exhaustion of the control groups.

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In a dual way, it can be operated on the basis of a partition by groups of variables instead of a partition by control groups

Figures 3, 4 and 5 schematically illustrate the message passing mechanism, respectively, for parallel planning, serial planning and mixed planning. It has been assumed that the controls were treated prior to the treatment of the variables. In Figure 3 the first stage of treatment of the controls (a) and the first stage of treatment of the variables (b) have been shown. In Figures 4 and 5 the first (a) and the second (c) stages of treatment of the controls have been shown, as well as the first stage (b) and the second stage (d) of treatment of the variables. In Figure 5, the set of control nodes has been divided into two groups indicated as G1 and G2 , which are treated sequentially.

It should be noted that serial planning and parallel planning, can in fact be seen as cases particular of a mixed planning, corresponding the first in case the groups are reduced to singletones (instances unique), and the second one corresponds to the case that a group understand the set of all control nodes (respectively of variables).

US 2005/0268204 describes a LDPC code decoder in which, based on the reliability  of the probability values of the binary elements acquired by the detection unit, a control unit determines a timing of operations to execute an operation LDPC

There are two main iterative decoding algorithms for message passing for LDPC codes: the SPA algorithm ( Sum Product Algoritm ), also called "log-BP", and the " Min-Sum " algorithm also called "based BP". A detailed description of these two algorithms can be found in the WE Ryan article entitled "An introduction to LDPC codes", published in "CRC Handbook for coding and signal processing for recording systems", and available at the link www.csee.wvu. edu / wcrl / ldpc.htm.

The SPA and Min-Sum algorithms only differ in the treatment stage of the controls that It will be detailed later. The other stages are identical, to to know:

The step 230 of treating the variables consists in calculating the messages α_ {mn} as follows:

8

where B 'mn represents the set of messages ? beta' mn received by the variable n from the controls m ' \ in H ( n ) - { m }, and C_ {mn} represents the event corresponding to a checked parity control for each of those controls. Subject to the independence of the y_ {n} , it is shown that α_ {mn} can be expressed in LLR form by:

9

The step 240 for estimating the information a posteriori consists in calculating:

10

in which B_ {n} represents the messages received by the variable n of all H ( n ) controls, and C_ {n} represents the event corresponding to a verified parity control for each of these controls. Under the same hypothesis as above, it is shown that \ hat {α} n can be expressed in the form of LLR by:

eleven

It is observed that, from (11) and (13):

12

The step 230 of treating the variables can therefore be located at the end of the iteration, after the estimation of the information afterwards . The expression (14) translates the fact that it does not return the extrinsic information (in this case \ beta_ {mn} ) sent by a node ( m ) to itself.

Stage 250 of decision of rigid values It is done simply by:

13

in which sgn ( x ) = 1 if x is positive, and sgn ( x ) = - 1 if it is not.

Verification in 260 of the controls of parity over rigid values, goes through the calculation of parity controls:

14

Parity controls are all satisfactory yes, and only if:

fifteen

The treatment stage of controls 220 It consists in calculating, using the SPA algorithm:

16

in which C_ {m} = 1 means a condition of parity satisfied for the control m , and A * _n} represents the set of messages \ alpha_ {mn '} received by the control m from part of the variables n' \ in H ( m ) - { n }. It is shown that β mn can be expressed by:

17

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being 18

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The treatment of controls according to Min-Sum algorithm corresponds to a simplification of the expression (19). Due to the rapid decrease of function \ Phi (x) and the equality of \ Phi (x) with its reciprocal, that is, \ Phi (\ Phi (x)) = x, you can legitimately make the following approach:

19

The decoding algorithm Min-Sum is significantly simpler than the algorithm of decoding SPA since it does not make more than additions, Comparisons and sign changes. In addition, the behaviors of Min-Sum algorithm are independent of the noise variance estimation sig2.

Although the algorithm returns from SPA decoding is better than the algorithm Min-Sum, they can be severely degraded in case of a bad estimate of the noise power.

The general object of the present invention it consists in proposing an iterative decoding algorithm of the type of message step, which allows you to decode a code error corrector that can be represented by graphic bipartite, that has better behaviors in terms of index of error and speed of convergence, that the algorithms of the same type known in the state of the art.

A first object of the invention consists in propose an iterative decoding algorithm of the step type of messages, which allows decoding an LDPC code with a complexity substantially lower than the SPA algorithm, all this presenting, for a signal-to-noise ratio given, comparable behaviors, or even higher, in error rate terms, and that, in addition, you do not need a noise power estimation.

Another particular object of the present invention is to propose a decoding algorithm of the type of message passing, which allows decoding of LDPC codes with a higher convergence rate than algorithms CPA or Min-Sum.

Exhibition of the invention

The present invention is defined by a iterative decoding method of message passing type to decode an error correction code susceptible to representation by means of a bipartite graph comprising a plurality of variable nodes and a plurality of nodes of control, said method being such that, for each iteration of a plurality of decoding iterations:

- a classification of the nodes of variable or control nodes depending on the degrees of respective reliability of decoding information available in the vicinity of these nodes, owning a node a high degree of reliability that is rated ahead of a node that has a low degree of reliability;

- the nodes so classified pass, each of they, at least one message to an adjacent node, in the order defined by the aforementioned classification.

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According to a first embodiment, said classification includes, for each node to be classified, the calculation of a measurement of reliability of the information present, sent or received by nodes located less than a distance default of the latter, within the bipartite chart, and the selection of the values of the measurements thus obtained.

For each iteration of the said plurality, then operates sequentially on the nodes classified in the order defined by the aforementioned classification and, for each node classified, messages are calculated for the nodes that they are adjacent to you and, for each of the aforementioned adjacent nodes, messages destined to the nodes adjacent to this one same.

According to a second embodiment, said classification includes, for each node to be classified, the calculation of a measurement of reliability of the information present, sent or received by nodes located less than a distance default of this node within the bipartite graph. The nodes are grouped by ranges of values of the aforementioned measurement.

If the reliability measurement has values integers, are stored for each of the aforementioned integer values, in an area of memory that is associated with this value, the indexes of the nodes whose reliability measurement is equal to the latter.

For each iteration of the said plurality, operates sequentially on the groups of nodes in the order defined by the aforementioned classification and, for each of the groups of nodes, the messages destined to the nodes are calculated they are adjacent and, for each of the aforementioned adjacent nodes, the messages destined to the nodes adjacent to these last.

According to a variant, each iteration of said plurality also includes, for each variable node of said bipartite graph, a step of calculating a priori information, already present in that node, and messages received by that node from the nodes adjacent control.

The step of calculating information a posteriori may be followed by a decision stage on the rigid value of said variable.

It is checked later if the values of the variables thus obtained verify the associated parity controls  to all control nodes of the graph and, if so, it provides as decoded word the word constituted by the aforementioned rigid values.

Advantageously, the classification is interrupted after a number of decoding iterations default, continuing the decoding procedure to continuation of your decoding iterations in the absence of the aforementioned classification of the aforementioned variable nodes or of the mentioned control nodes.

The classification is interrupted if the minimum, in absolute value, of the differences between the a posteriori values and the a priori values of the aforementioned variables, is greater than a predetermined threshold value, continuing the decoding procedure in the absence of said classification of the aforementioned variable nodes or the mentioned control nodes.

In a first application, the aforementioned code Bug fixer is a turbocode.

In a second application, said error correction code is an LDPC code ( K, N ) represented by a bipartite graph of N variable nodes, and M = NK control nodes.

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In the latter case, the message \ betamn of an index control node m \ in {1, ..., M ) to an index variable node n \ in {1, ..., N ), may be calculated by:

twenty

wherein \ alpha_ {mn '} designates the message from the index variable node n ' to the index control node m , H ( m ) the set of variable nodes adjacent to the index control node m , sgn ( x ) = 1 if x is positive and sgn ( x ) = - 1 if it is not,

Y twenty-one

Alternatively, the message \ beta_ {mn} from an index control node m \ in {1, ..., M } to an index variable node n \ in {1, ..., N ) can be calculated through:

22

wherein \ alpha_ {mn '} designates the message from index variable node n ' to index control node m , H ( m ) the set of variable nodes adjacent to index control node m , sgn ( x ) = 1 if x is positive and sgn ( x ) = - 1 if it is not.

According to an embodiment, the classification falls on the control nodes, and the predetermined distance is equal to 2. The reliability measurement of an index control node m is then calculated as:

2. 3

wherein H ( m ) designates the set of variable nodes adjacent to the index control node m , H ( n ) the set of control nodes adjacent to the index variable node n , Card (.) designates the cardinal of a set, with c = 0 if c_ {m} = + 1 and c = \ delta_ {max} +1 if c_ {m} = -1, where \ delta_ {max} is the maximum degree of the nodes of control in the bipartite graph, and in which c_ {m} = + 1 / c_ {m} = -1 mean, respectively, that the parity control has been verified / has not been verified, for the index control node m .

The invention relates, finally, to a computer program comprising logical means adapted to carry out the steps of the decoding procedure defined previously when it is executed by a computer.

Brief description of the drawings

Figure 1 represents an example of the Tanner chart for a code ( K, N );

Figure 2 illustrates, schematically, the principle of iterative decoding per message step known in the state of the art;

Figure 3 represents the first iterations of an iterative decoding in message step according to a parallel type planning;

Figure 4 represents the first iterations of an iterative decoding in message step according to a serial type planning;

Figure 5 represents the first iterations of an iterative decoding in message step according to a mixed type planning;

Figures 6A and 6B represent, respectively, the neighborhood of order 1 and order 2 of a node;

Figure 7 represents the beginning of a iterative decoding in message step according to the invention;

Figure 8 represents a procedure of iterative decoding in message step according to a first mode of embodiment of the invention;

Figure 9 represents a procedure of iterative decoding in message passing according to a second mode of embodiment of the invention;

Figure 10 represents an application example of the decoding process according to the invention, for decode an LDPC code, and

Figure 11 indicates the error rate obtained by the decoding process according to the invention by comparison with prior art methods, for a code LDPC particular.

Detailed exposition of particular embodiments

Again, a corrective error code is considered susceptible to a representation by means of a bipartite graph with N variables and M controls.

The distance between two nodes \ nu, \ mu of the graph is called and is indicated as D ( \ nu, \ mu ), the length, expressed in number of edges, of the shortest path that joins these nodes through the graph. Taking into account the very definition of a bipartite graph, it follows that the distance between two nodes of the same type is an even number, and that the distance between two nodes of different types is an odd number.

The neighborhood of order d of any node \ nu of a graph \ Gamma is defined, the set V _ {\ nu} ^ {(d)} of the nodes located at a distance less than or equal to d from \ nu , that is :

24

Thus, the neighborhood of order 0 of a node is none other than the node itself, the neighborhood of order 1 is the meeting of the neighborhood of order 0 with the set of nodes adjacent to that node, the neighborhood of order 2 is the meeting of the order 2 neighborhood of order 1 with the set of nodes adjacent all around the nodes of the latter, and so on. For a bipartite graph, consisting of variables n and controls m , the neighbors of order 0 to 2 of these nodes are given by:

25

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It will be appreciated that the singlet {n} (respectively {m}) does not appear in the expression V_ {n} ^ {(2)} (respectively V_ {m} ^ {2)} because it is already included in the second term of the union taking into account the symmetry of the adjacency relationship.

In general, the neighbors of a node \ nu of a bipartite graph can be obtained through the recurrence relationship:

26

In order to explain the notion of neighborliness, the neighbors of order 1 and 2 of a control node m of the graph of Figure 1 have been represented in Figures 6A and 6B. The nodes belonging to said neighborhoods have been indicated in crosslinked

Hereinafter, measurement reliability of order d of a \ nu \ Gamma a variable f _ {d} (\ nu) node dependent information decoding available within the vecinaje V _ {\ Gamma} is called ^ (d)} , that is, of the information present, sent or received by the nodes belonging to said neighbor, and which indicate the degree of local reliability of the decoding operation. Here we will rate the measurement as "positive" when f d (nu) is higher than the degree of reliability, and as "negative" otherwise.

The decoding information available on a control node m comprises, on the one hand, the value c_ {m} , which indicates the verification or not of the parity control and, on the other hand, the messages ? Β {mn} transmitted to the adjacent variable nodes n \ in H ( m ). The value c_ {m} , as well as the messages \ beta_ {mn} are carriers of reliable information about decoding.

The data available in a node n variable comprising the information a priori \ alpha {n}, the a posteriori information \ hat {\ alpha} _ {n}, messages extrinsic information \ alpha {mn} destined to nodes adjacent control m \ in H ( n ), as well as the rigid values \ overline {\ alpha} _ {n}.

The information cited above, with the exception of rigid values, they are carriers of a Reliability information about decoding.

In general, if the flexible values ? Beta_ {mn}, \ alpha_ {mn}, \ alpha_ {n} , \ hat {\ alpha} {n}, expressed in the form of LLR are considered, decoding will be considered both more reliable the higher in absolute value the latter are.

In the following we will give some examples illustrative and non-limiting reliability measurements for Orders 0 to 2:

- reliability measurements of order 0:

The neighborhood is then reduced to the node itself. The function can be chosen for the measurement of zero order reliability of a node n of variable:

27

which indicates the degree of confidence in rigid value \ overline {\ alpha} _ {n}.

For a control node m you can simply choose the function:

28

It is understood, in effect, that if it is verified that the parity control has been ( c_ {m} = 1), the decoding will be more reliable than if it is not verified.

- reliability measurements of order 1:

Variable node and node will be distinguished again of control. For a variable node you can choose for the reliability measurement of order 1, one of the functions following:

29

30

The expression (27) represents the number of control nodes m linked to the variable node n for which the parity control is verified.

The expression (28) takes into account the irregularity of the code (i.e. the possibility of nodes variable have different degrees), and takes stock of Verified and unverified parity controls.

The expressions (27 ') and (28') are measurements negative corresponding to the positive measurements (27) and (28).

The measurement of reliability expressed in (29) is conservative, since it is founded on a reduction of the reliability information received from the control nodes adjacent.

For a control node m , one of the following functions may be chosen for reliability measurement of order 1:

31

The expression (30) indicates the minimum of the reliability (information a posteriori ) of the variables to which the control m is connected.

The expression (31) translates a similar criterion, but it falls on extrinsic information. It will be appreciated that the It is dual of the expression (29).

The expression (32) represents the Cartesian product of two information and, therefore, they are taken into account jointly: the first falls on the verification of the parity control and the second on the subsequent information, as in (30).

- reliability measurements of order 2:

The following function can be chosen for a variable node n :

32

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The expression (33) represents a negative measurement of decoding reliability: it indicates the number of controls connected to the variable n for which the parity condition has not been verified, and such that the variable n is the least reliable among all variables adjacent to these controls. In fact, measurement (33) can be considered as the composition of measurements (27 ') and (30).

For a control node m , one of the following functions can be chosen:

33

with c = 0 if c_ {m} = + 1 and c_ {m} = \ delta_ {max} +1 if c_ {m} = -1 where \ delta_ {max} is the maximum degree of the control nodes, that is : 104

The expression (34) provides the number of unverified parity controls in the neighborhood of order 2 of m , excluding m itself. This represents a negative measure of the decoding reliability. It will be appreciated that the second equality of the expression (34) is verified only in the absence of cycles of length 4 in the neighborhood in question. In practice, codes whose Tanner graphics have cycles as long as possible are used, so that equality is generally verified.

The expression (35) indicates the number of variables adjacent to m , connected to at least one unverified control, excluding m itself. It therefore represents a negative measure of the reliability of decoding. Since the terms under the sum sign are equal to 0 or 1, the measurement value is between 0 and δ_max .

The expression (35) makes no distinction as to whether or not the parity control has been verified for node m . In order to take this information into account, an interpolation c whose value depends on the verification of the control m has been introduced in expression (36). For the interpolation values indicated above ( c = 0, δ_ {max} +1), the reliability measurement takes values between 0 and δ_max for a verified control, and between \ delta_ {max} + 1 and 2 \ delta_ {max} +1 for an unverified control. We note that this is a negative reliability measurement in the sense defined above, that is, the more reliable the value of \ tilde {f} 2 (m), the more reliable the control m . Thus, the degree of reliability of a verified control is always higher than that of an unverified control, whatever the value of (35).

It will be appreciated that some of the functions defined above are integer values, in particular the from (34) to (36). They lend themselves particularly well to a implementation through a microprocessor.

Importantly, some of the functions defined above, in particular those from (34) to (36), refer only to rigid values c_ {m} . Since 105 , it is noted that the value of the noise power ( sig2 ) does not intervene in the calculation of these functions.

The idea that underlies the basis of the invention is to classify the nodes of the graph according to their reliability, and perform a decoding by passing messages according to a scheme of serial or mixed type starting with the node or group of nodes more reliable (s). This improves the behavior considerably of decoding, both in terms of speed of convergence as binary error rate (BER).

It is understood that the messages transmitted by the most reliable nodes are the best to contribute effectively to decoding of adjacent nodes. There is thus a rapid propagation of reliability from one iteration to another and node in node, and correlatively, a sensible increase in value absolute of extrinsic information, tailored to the iterations

Figure 7 schematically illustrates the decoding principle according to the invention. After an initialization phase 710 that will be detailed later, an iteration loop is entered. In 720, the classification of the control nodes or the variable nodes of the graph is determined according to their degree of reliability. It will be assumed here that the control nodes are classified, leading the classification of the variable nodes to similar stages.
Tricas

In 730, a serial treatment is carried out or a mixed treatment For a series treatment, they are treated control nodes sequentially and, for each of them, the variable nodes that are adjacent to it, starting with the node of more reliable control and continuing through degrees of reliability decreasing For a mixed type treatment, the nodes of control will have been classified into reliability groups, being the first group consisting of the most reliable nodes, being the second group consisting of less reliable nodes, etc. It is about then, in parallel, the variable nodes and the nodes of control of the first group, and passed immediately after parallel treatment of the variable nodes and the nodes of control of the next group, and it continues like this until exhaustion of the groups. Stages 740, 750, 760, 765, 767, 770, 775 are, respectively, identical to steps 240, 250, 260, 265, 267, 270, 275 of Figure 2, and therefore no detail.

Figure 8 represents, in more detail, an embodiment of the invention that uses planning  Serie.

In step 810, an initialization of the messages α_mn and βmn is performed . For example, for each pair of variable n and control m \ in H ( n ), \ alpha_ {mn} is initialized by α_ {mn} = \ alpha_ {n} . Similarly, for each control pair m and variable n \ in H ( m ), β_mn} is initialized using β_ {mn} = 0.

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The controls are then classified in 820, by decreasing degrees of reliability using the reliability measurements \ tilde {f} d (m). In other words, for a positive measurement, controls are classified so that 106 and for a negative measurement they are classified in the opposite direction. If the measurement is a Cartesian product (see, for example, (32)), the inverse lexicographic order relationship is used according to the negative or positive measurement. On the other hand, a control node counter, j = 0, is initialized.

The current control m = m_ {j} is selected in 825 and the variables n \ in H ( m ) are treated in 830, or in other words it is calculated 107

Then, in 835, a treatment of the current control m is passed, or in other words it is calculated 108 for n \ in H ( m ).

According to a variant embodiment, they are reversed stages 830 and 935.

The counter of control nodes is increased by 837, and it is checked at 939 if they have been treated in their entirety. If not, return to step 825, and if not, follow steps 840 to 860, which respectively represent the calculation of information a posteriori , the decision on rigid values and the verification of parity controls . Steps 840, 850, 860, 865, 867, 870, 875 are respectively identical to steps 240, 250, 260, 265, 267, 270, 275 of Figure 2.

According to an alternative embodiment, proceed to classify the variables instead of the classification of controls. The internal loop 825 to 839 carries about one variable at a time, the same as in the foregoing, being able to carry out the treatment of the current variable before or after the treatment of the controls that are adjacent to it.

Figure 9 illustrates an embodiment of the invention that uses mixed planning.

In step 910, an initialization of the messages α_ {mn} and β_ {mn} is carried out as in step 810 of Figure 8.

In 920, the set of control nodes or variable nodes is divided into groups, each corresponding to a different reliability class. A reliability class is defined, for example, by a range of values of a reliability measurement. This embodiment is particularly advantageous when the measurement is of integer values. In effect, you can then assign a group to an integer value or a plurality of continuous integer values, and order the indexes of the nodes in a table according to their respective classes. It will be appreciated in what follows by \ varphi_ {0}, \ varphi_ {1}, ...., \ varphi_ {\ omega} the different reliability classes, with \ varphi_ {0} being the highest degree of reliability class and var the lowest degree of reliability class, and G 0, G 1, ...., G the groups of corresponding nodes. It is assumed, in the following, that the partition has been effected on the control nodes.

On the other hand, the reliability class counter is initialized at j = 0.

In 925, the group of nodes corresponding to the current reliability class \ varphi_ {j} , that is G_ {j}, is selected .

In step 930, the treatment of the variables n \ in H ( m ), \ m \ m \ in G_ {j} is carried out in parallel, that is, the messages are calculated 109

In step 935, the processing of the controls m \ in G_ {j} is carried out in parallel, that is, the messages are calculated 110

According to a variant embodiment, they are reversed stages 930 and 935.

In 937 the class counter is increased and it is checked, in 939, if all of them have been treated. If not, it returns to the loop in step 925, if not, it continues until steps 940 to 960, which respectively represent the calculation of information afterwards , the decision on rigid values and the verification of parity controls . Stages 940, 950, 960, 965, 967, 970, 975, are respectively identical to steps 240, 250, 260, 265, 267, 270, 275 of Figure 2.

Figure 10 illustrates a particular application of the invention to decoding an LDPC code by means of mixed planning, based on a measurement of reliability to integer values. The negative measurement on the control nodes defined in (36).

In step 1010, the initialization of the flexible values is carried out a posteriori \ hat {\ alpha} {n} by means of the corresponding observations: = \ hat {\ alpha} {n} = \ alpha_ {n} and of the messages? mn for 0.

The counter is also initialized iterations to 0.

In 1015, a decision is made about the values rigid, that is \ overline {\ alpha} _ {n} = sgn (\ hat {\ alpha} _ {n}).

The value of 1020 is calculated 3. 4 in which the c_ {m} are the parity controls 35 350

It is checked, in 1023, if \ chi = M, that is, if all the parity controls have been verified. If so, the decoding algorithm is exited in 1025, providing the code word. Otherwise, it is continued in 1027 by calculating the reliability measurements of the control nodes by means of the measurement (36). Remember that this measurement takes integer values between 0.1, ..., 2 \ delta_ {max} +1, where \ delta_ {max} is the maximum degree of control nodes. In the present case, a reliability class is assigned by integer value, that is, \ varphi_ {0,} \ varphi_ {1}, ..., \ varphi_ {\ omega} with \ omega = 2 \ delta_ {max} +1 in which the class \ varphi_ {j} corresponds to the measurement value j \ in {0,1, ..., 2 \ delta_ {max} +1}.

In 1029, the group of nodes corresponding to the current reliability class \ varphi_ {j} is selected, ie G_ {j} .

In 1030, the treatment of the variables n \ in H ( m ), \ forall m \ inG_ {j} is carried out in parallel, that is, the messages \ alpha_ {mn} = \ hat {\ alpha} _ { n} - \ beta_ {mn} , \ forall m \ inG_ {j}.

In step 1035, the processing of the controls m \ inG_ {j} is carried out in parallel, that is to say that the messages with destination to the variables n \ in H ( m ), \ forall m \ inG_ {j} are calculated, that is to say,

36

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for a SPA treatment Y

37

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for a simplified Min-Sum type treatment as indicated herein on the Figure. They are then calculated a posteriori information, in 1040, thanks to 38 .

It will be appreciated that a part of the controls m \ in H ( n ) involved in the sum, may not be part of the group G_ {j} and therefore the messages \ beta_ {mn} that come from those controls have not been recalculated at 1035. In addition, since the variable n is adjacent to several controls m \ in H ( n ), the calculation of \ hat {\ alpha} n will be done several times as the controls of H ( n ). In order to avoid that the sum in 1040 is not calculated several times and that every time some [beta] {mn} messages have not changed, they are not recalculated, steps 1035 and 1040 can be advantageously replaced by a stage that includes the following operations:

\ forall m \ inG_ {j} , \ forall n \ in H ( m ),

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subtraction of the old message, is tell:

\ hat {\ alpha} n = \ hat {\ alpha} _ {n} - \ beta_ {mn}

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calculation of the new message, is tell:

103

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update the information a posteriori by adding the new message:

\ hat {\ alpha} _ {n} = \ hat {\ alpha} _ {n} + \ beta_ {mn}

In this way, the information is updated a posteriori \ hat {\ alpha} {n} as the new messages sent by the controls m \ in H ( n ) to the variable n are calculated. This makes it possible to significantly reduce the number of operations carried out for the calculation of the information afterwards .

Then it increases, by 1043, the reliability class counter and it is checked, in 1045, if all The classes have been treated. If not, return to the stage 1029 to address the following reliability class. In case Yes, the iteration counter is incremented by 1050. In 1053, it is checked if the detention criterion is verified, it is say, if the maximum number of iterations has been reached. Yes it is In that case, the loop is released in 1055, concluding the failure of the decoding. By default, the loop is returned in step 1015 For a new iteration.

It can be seen that in Figure 10, the decision stages are the rigid values, both of verification of the parity control prior to those of calculation of reliability, and of treatment of the nodes. This is due to the fact that the reliability measurement used here (expression (36)), resorts to the rigid decisions c_ {m} and therefore needs its prior calculation.

The algorithm illustrated in Figure 10 does not it presents more than a low additional complexity in relation to SPA or Min-Sum algorithms of the prior art. In return, neighborhood reliability planning accelerates convergence and allows to reach an error rate (BER) lower for the same signal-to-noise ratio (SNR). Alternatively, for the same setpoint error rate, the decoding process according to the invention allows use codes at a lower coding gain.

Figure 11 allows comparing behaviors, in terms of binary error rate (BER), of the conventional Min-Sum and SPA algorithms, with the Min-Sum algorithm using a schedule by neighborhood reliability, indicated as Min-Sum-FV, all for a range Signal-to-noise ratio important. The codes used are irregular LDPC codes [1008,2016] and [4032,8064], and therefore 1/2 performance. The number of iterations performed It is equal to 200 for all three algorithms.

It is observed that the algorithm Min-Sum-FV according to the invention, surpasses the conventional SPA algorithm in the sense that its complexity is significantly less.

In the decoding process according to the invention such as that shown in Figures 7 to 10, it has been assumed that neighborhood reliability planning was used for the entire duration of the decoding. According to a variant, this planning cannot be applied except during the first iterations, in order to eliminate the noise of the observations, and then to tilt over a conventional treatment. The tilting criterion can be labeled in a number of iterations Iter_ {f} or, advantageously, based on an absolute value comparison of the extrinsic information with a reference threshold. For example, after iterative decoding:

39

where T_ {f} represents a minimum reliability threshold, a conventional Min-Sum or SPA treatment can be returned. This variant allows to avoid the selection or classification of the nodes after the decoded values are sufficiently reliable and the convergence is assured.

The present invention applies to the decoding of error correction codes susceptible to representation by bipartite graph, especially LDPC codes or turbo codes. It can be used in the field of data logging or telecommunications, in particular for telecommunications systems that already use LDPC codes, for example those that comply with IEEE standards 802.3a (Ethernet 10 Gbits / s), DVB-S2 (video broadcast over satellite), IEEE 802.16 (WiMAX), or are likely to use them, for example systems that comply with IEEE 802.11 standards (WLAN) and IEEE 802.20 (Mobile Broadcasting Wireless Access).

Claims (17)

1. Iterative decoding procedure of the type of message step to decode an error correction code capable of being represented by a bipartite graph comprising a plurality of variable nodes and a plurality of control nodes, characterized in that, in each iteration of a plurality of decoding iterations: - a classification (720) of the variable nodes or of the control nodes is made according to the respective degrees of reliability of the decoding information available in the neighborhoods ( V_ {n} ^ {(d)} , V_ {m} ^ (d)} ) of these nodes, so that a node that has a high degree of reliability is classified ahead of a node that has a low degree of reliability; - the nodes so classified pass (725), each of them, at least one message ( α_ {mn}, \ beta_ {mn} ) to an adjacent node, in the order defined by said classification. 2. Decoding method according to claim 1, characterized in that said classification comprises, for each node to be classified, the calculation of a measurement of reliability ( f d ( n ), (\ tilde {f} d) (m)) of the information present, sent or received by the nodes located less than a predetermined distance ( d ) of the latter within the bipartite graph, and the selection of the values of the measurements thus obtained. 3. Decoding method according to claim 2, characterized in that, for each iteration of said plurality, it operates sequentially on the nodes classified in the order defined by said classification, and because, for each classified node (825), the messages destined to the nodes that are adjacent to it (835) and, for each of the aforementioned adjacent nodes, the messages destined to the nodes adjacent to it (830). 4. Decoding method according to claim 1, characterized in that said classification comprises, for each node to be classified, the calculation of a measurement of reliability ( f d ( n ), (\ tilde {f} d) (m)) of the information present, sent or received by the nodes located less than a predetermined distance ( d ) from this node within the bipartite graph, and because the nodes are grouped by intervals of values of the said measurement. 5. Decoding method according to claim 4, characterized in that the measurement of reliability is in integer values, and because, for each of said integer values, the indices of the memory are associated with them. nodes whose reliability measurement is equal to that value. 6. Decoding method according to claim 4 or 5, characterized in that, for each iteration of said plurality, the groups of nodes are sequentially operated in the order defined by said classification, and because, for each group of nodes ( G_ {j} ), the messages destined to the adjacent nodes (935) are calculated and, for each of the aforementioned adjacent nodes, the messages destined to the nodes adjacent to the latter nodes (930). 7. Decoding method according to one of the preceding claims, characterized in that each iteration of said plurality further comprises, for each variable node of said bipartite graph, a calculation step (740, 840, 940) of a posteriori information (\ hat {\ alpha} _ {n}) based on a priori information (\ alpha_ {n}) already present in that node and the messages received ( \ beta_ {mn} ) by that node from the nodes adjacent control. 8. Decoding method according to claim 7, characterized in that the step of calculating information a posteriori is followed by a decision stage (750, 850, 950) on the rigid value (\ hat {\ n}) of the aforementioned variable. 9. Decoding method according to claim 8, characterized in that it is checked (760, 860, 960) if the rigid values of the variables thus obtained verify the parity controls associated with all control nodes of the graph, and why, in If so, the word constituted by the said rigid values is provided as a decoded word. 10. Decoding method according to one of the preceding claims, characterized in that the classification is interrupted after a predetermined number of iterations of decoding ( Iter f), continuing the decoding procedure following its decoding iterations in the absence of the cited classification of said variable nodes or of said control nodes. 11. Decoding method according to claim 7, characterized in that the classification is interrupted if the minimum, in absolute value, of the differences between the a posteriori values and the a priori values of said variables, is greater than a predetermined threshold value ( T f), the decoding procedure then continuing its decoding iterations in the absence of said classification of said variable nodes or of said control nodes.

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12. Decoding method according to one of the preceding claims, characterized in that said error correction code is a turbocode. 13. Decoding method according to one of claims 1 to 11, characterized in that said error correction code is an LDPC code ( K, N ) represented by a two-part graph of N variable nodes and M = N - K control nodes . 14. The decoding method according to claim 13, characterized in that the message \ beta_ {mn} of an index control node m \ in {1, .., M } to an index variable node n \ in {1, .., N }, is calculated by: 40 wherein \ alpha_ {mn '} designates the message of the index variable node n' to the index control node m, H ( m ) the set of variable nodes adjacent to the index control node m , sgn ( x ) = 1 if x is positive and sgn ( x ) = - 1 if it is not, and 41 15. Decoding method according to claim 13, characterized in that the message \ beta_ {mn} of an index control node m \ in {1, .., M } to an index variable node n \ in {1, .., N }, is calculated by: 42 wherein \ alpha_ {mn '} designates the message of the index variable node n' to the index control node m , H ( m ) the set of variable nodes adjacent to the index control node m , sgn ( x ) = 1 if x is positive, and sgn ( x ) = - 1 if it is not. 16. Decoding method according to claims 5 and 15, characterized in that the classification falls on the control nodes, because said predetermined distance is equal to 2, and because the reliability measurement of an index control node m is calculated by : 43 wherein H ( m ) designates the set of variable nodes adjacent to the index control node m , H ( n ) the set of control nodes adjacent to the index variable node n , Card (.) designates the cardinal of a set, with c = 0 if c_ {m} = + 1 and c = \ delta_ {max} +1 if c_ {m} = -1 in which \ delta_ {max} is the maximum degree of the nodes of control in the bipartite graph, and in which c_ {m} = + 1 / c_ {m} = -1 mean respectively that the parity control has been verified / has not been verified, for the index control node m . 17. Computer program comprising media logics adapted to carry out the steps of the procedure decoding according to one of the preceding claims, when it is executed by a computer.